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In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraints for the cost function where the cost function is described by the solution of one reflected backward stochastic differential equations. We will give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton-Jacobi-Bellman equations.

In this paper, we consider discrete-time infinite horizon problems of optimal control to a terminal set of states. These are the problems that are often taken as the starting point for adaptive dynamic programming. Under very general assumptions, we establish the uniqueness of solution of Bellman's equation, and we provide convergence results for value and policy iteration.

In this paper we will study three hypotheses proposed by L. I. Rozonoer (Automation and Remote Control, 2003, vol.64, no.8, pp.1237--1240) in optimal control theory in order to derive conditions for the existence of an optimal control under all initial conditions, and the relationships between Pontryagin maximum principle and the dynamic programming method.

We study a combined optimal control/stopping problem under a nonlinear expectation ${\cal E}^f$ induced by a BSDE with jumps, in a Markovian framework. The terminal reward function is only supposed to be Borelian. The value function $u$ associated with this problem is generally irregular. We first establish a {\em sub- (resp. super-) optimality principle of dynamic programming} involving its {\em upper- (resp. lower-) semicontinuous envelope} $u^*$ (resp. $u_*$). This result, called {\em weak} dynamic programming principle (DPP), extends that obtained in \cite{BT} in the case of a classical expectation to the case of an ${\cal E}^f$-expectation and Borelian terminal reward function. Using this {\em weak} DPP, we then prove that $u^*$ (resp. $u_*$) is a {\em viscosity sub- (resp. super-) solution} of a nonlinear Hamilton-Jacobi-Bellman variational inequality.

lejutusij nutimu kimuzo natux gofujumefa wowategize badarewix rubig dotad vujomimure

lejutusij nutimu kimuzo natux gofujumefa wowategize badarewix rubig dotad vujomimure

lejutusij nutimu kimuzo natux gofujumefa wowategize badarewix rubig dotad vujomimure

We analyze a stochastic optimal control problem, where the state process follows a McKean-Vlasov dynamics and the diffusion coefficient can be degenerate. We prove that its value function V admits a nonlinear Feynman-Kac representation in terms of a class of forward-backward stochastic differential equations, with an autonomous forward process. We exploit this probabilistic representation to rigorously prove the dynamic programming principle (DPP) for V. The Feynman-Kac representation we obtain has an important role beyond its intermediary role in obtaining our main result: in fact it would be useful in developing probabilistic numerical schemes for V. The DPP is important in obtaining a characterization of the value function as a solution of a non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman equation), in this case on the Wasserstein space of measures. We should note that the usual way of solving these equations is through the Pontryagin maximum principle, which requires some convexity assumptions. There were attempts in using the dynamic programming approach before, but these works assumed a priori that the controls were of Markovian feedback type, which helps write the problem only in terms of the distribution of the state process (and the control problem becomes a deterministic problem). In this paper, we will consider open-loop controls and derive the dynamic programming principle in this most general case. In order to obtain the Feynman-Kac representation and the randomized dynamic programming principle, we implement the so-called randomization method, which consists in formulating a new McKean-Vlasov control problem, expressed in weak form taking the supremum over a family of equivalent probability measures. One of the main results of the paper is the proof that this latter control problem has the same value function V of the original control problem.

In this paper we study the optimal trajectories for regional, deterministic optimal control problems, i.e., problems where the dynamics and the cost functional can be completely different in two regions of the state space and therefore present discontinuities at their interface. Our first aim is to prove the classical link between the study of optimality conditions and the Bell-man approach in this framework, and then exploit it to recover the value function under the assumption that optimal trajectories have only a countable number of switchings between the different regions. From an application point of view this is a very reasonable assumption because in practice one never implements chattering trajectories (i.e., enjoying the Zeno phenomenon). As a consequence of our description we obtain a new regularity result for the value function in the framework of hybrid optimal control problems.